Non-uniqueness of Admissible Weak Solutions to Compressible Euler Systems with Source Terms

TL;DR

This paper demonstrates the non-uniqueness of admissible weak solutions to the compressible Euler system with various source terms, including rotation and damping, using convex integration and localized plane wave perturbations.

Contribution

It extends convex integration techniques to Euler systems with source terms, constructing infinitely many solutions under broad conditions.

Findings

Infinitely many global admissible weak solutions for anti-symmetric sources.

Construction of finite-states admissible solutions with anti-symmetric sources.

Multiple solutions exist under small initial density assumptions for general sources.

Abstract

We consider admissible weak solutions to the compressible Euler system with source terms, which include rotating shallow water system and the Euler system with damping as special examples. In the case of anti-symmetric sources such as rotations, for general piecewise Lipschitz initial densities and some suitably constructed initial momentum, we obtain infinitely many global admissible weak solutions. Furthermore, we construct a class of finite-states admissible weak solutions to the Euler system with anti-symmetric sources. Under the additional smallness assumption on the initial densities, we also obtain multiple global-in-time admissible weak solutions for more general sources including damping. The basic framework are based on the convex integration method developed by De~Lellis and Sz\'{e}kelyhidi \cite{dLSz1,dLSz2} for the Euler system. One of the main ingredients of this paper is the construction of specified localized plane wave perturbations which are compatible with a given source term.